Tori, invariant

This shape change may be related to the properties of the invariant tori by use of the tangent relation [21, 22]... 

The nature of the above distortion leads to an associated multivaluedness in the vibrational quantum number, which can be attributed to the fact that the twist angle of the classical invariant tori increases 2n on a cycle around the critical singularity. 

A. Jorba and J. Villanueva, On the persistence of lower dimensional invariant tori under quasi-periodic perturbations, J. Nonlinear Sci. 7, 427 (1997). 

In the present paper we study common features of the responses of chemical reactor models to periodic forcing, and we consider accurate methods that can be used in this task. In particular, we describe an algorithm for the numerical computation and stability analysis of invariant tori. We shall consider phenomena that appear in a broad class of forced systems and illustrate them through several chemical reactor models, with emphasis on the forcing of spontaneously oscillating systems. 

FIGURE 10 Change of the torus and the angular function with W o. (a) A succession of computed sections of invariant tori for various values of >/ >o (Brusselator, a = 0.0072). The centre point is indicated by (+). (b, c) The occupancy of the converged Jacobian for (u/gjo = 1.186667 and 1.3, respectively. The bumps on some of the circles are artifacts of the mesh and are associated with the almost vertical parts of the nonzero band. They can be eliminated by mesh adaptation. 

Thoulouze-Pratt, E., 1983, Numerical analysis of the behaviour of an almost periodic solution to a periodic differential equation, an example of successive bifurcations of invariant tori. In Rhythms in Biology and Other Fields of Application, Lect. Notes in Biomath, Vol. 49, pp. 265-271. 

A complete model for the non-ergodic classical dynamics of a polyatomic molecule will need to represent the complete Arnold web structure of the phase space. There may be multiple bottlenecks for IVR and vague tori may exist in the vicinity of invariant tori. These complex phase space structures, leading to non-ergodic dynamics, are the origins of the... 

In addition to computer simulations, what drives the research in this direction is elaborated perturbation theories developed almost simultaneously. In particular, the Kolmogorov-Arnold-Moser (KAM) theorem, which has shown the existence of invariant tori under a small perturbation to completely inte-grable systems, and the Nekhoroshev theorem, which has proved exponentially long-time stability of trajectories close to completely integrable ones, are landmarks in this field. Although a lot of works have been done, there still remain unsolved important questions, and the Hamiltonian system is being studied as one of important branches in the theory of dynamical systems [3-5]. 

On the other hand, if the perturbation theory is applied to the completely integrable system, the existence of invariant tori can be discussed. More precisely, consider iV-degrees-of-freedom Hamiltonians under a small perturbation in the standard action-angle form,... 

In addition, we should remark that invariant tori we often found in numerical simulations are not truly KAM tori guaranteed rigorously in the mathematical theorem. The perturbation strength s is so small and any chaotic orbits cannot be detected in phase space if we perform numerical simulations under the original condition of the KAM theorem as for the smallness of . 

In contrast to hyperbolic systems, the phase space structure in the mixed system is quite intricate and inhomogeneous, which brings about transport phenomena and relaxation processes essentially different from uniformly hyperbolic cases [3]. A remarkable fact is that qualitatively different classes of motions such as quasi-periodic motions on invariant tori and stochastic motions in chaotic seas coexist in a single phase space. The ordered motions associated with invariant tori are embedded in disordered motions in a self-similar way. The geometry of phase space then reflects the dynamics. 

If the resonant tori, which are the invariant tori whose rotational numbers are rational, are broken under perturbations, the pairs of elliptic and hyperbolic cycles are created in the resonance zone. This fact is known as a result of the Poincare-Birkhoff theorem [4], which holds only if the twist condition, Eq. (2), is satisfied. Around elliptic cycles thus created, new types of tori, which are... 

Figure 1. Composite of several Polncar6 surfaces of section for the mass ratio Pi2 m2 = 1 64 (appropriate to H2O), at the classical dissociation energy, D> Invariant tori dominate the phase space structure Reproduced from ref (16), with permission. Copyright 1983, American Institute of Physics.

The results of this section show that semlclasslcal estimates of resonance energies may be obtained — using the Solov ev—Johnson ansatz In regions of phase space where Invariant tori do not exist. This technique may well turn out to be the semlclasslcal method of choice for many-degrees-of-freedom systems even If tori exist, visualization of the caustic structure Is difficult. 

Thus, the orbits in the domain Q x Tn of phase space lie on invariant tori parameterized by the action variables h,..., /n, and the motion on each torus is a Kronecker flow with frequencies wi(/),..., ojn(I). 

This section is devoted to the statement of the theorem of Kolmogorov on the persistence of Invariant tori. We first state the theorem and recall in some detail the formal method invented by Kolmogorov. Then we show how the original scheme can be rearranged in the form of a constructive algorithm, suitable for an explicit calculation via algebraic manipulations. Finally we shall apply this method to the problem of three bodies. 

A first application of the normal form method is the proof of the theorem of Kolmogorov on the persistence of invariant tori. We outline here the basic scheme proposed in Kolmogorov s original paper. 

The conclusion is that at least in the approximation of the secular system the orbit of Jupiter and Saturn is confined forever between two KAM invariant tori, thus assuring the perpetual stability of this system. Below we resume in brief the method and the result. For more details the reader is referred to Locatelli and Giorgilli (2000). 

Figure 3. Illustrating the topological confinement of the orbit in the 4D phase space. The continuous curves T and Y" represent two sets of 2D invariant tori that intersect transversally an energy surface. An orbit with initial datum in the gap between two tori will be eternally trapped in the same region (see text).

The procedure above allows us to prove only that there is an invariant torus close to the initial conditions of the Sun-Jupiter-Saturn system, not that the orbit of the system actually lies on a torus. Since we can not exclude the possibility of Arnold s diffusion, this is not enough to prove the perpetual stability of the orbit of the secular system. Therefore, we make a more accurate analysis in order to prove that the orbit is actually confined in a gap between two invariant tori. The procedure is illustrated in Figure 3. 

Benettin, G., Galgani, L., Giorgilli, A. and Strelcyn, J. M. (1984). A proof of Kolmogorov s theorem on invariant tori using canonical transformations defined by the Lie method. R Nuovo Cimento, 79 201-216. 

Giorgilli, A. and Locatelli, U. (1999). A classical self-contained proof of Kolmogorov s theorem on invariant tori. In Simo, C., editor. Hamiltonian systems with three or more degrees of freedom, NATO ASI series C, 533. Kluwer Academic Publishers, Dordrecht-Boston-London. 

Locatelli, U. and Giorgilli, A. (2000). Invariant tori in the secular motions of the three-body planetary systems. Cel. Mech., 78 47-74. 

According to Nekhoroshev (1977) and to Morbidelli and Giorgilli (1995), the old and crucial question of stability of a dynamical system turns out to be related to the structure and density of invariant tori which foliate the phase space. For instance the puzzle of the 2/1 gap of the asteroidal belt distribution was explained showing that the corresponding region of the phase space is a weak chaotic one (Nesvorny and Ferraz-Mello 1997). 

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